The Canaries were hammered 4-0 at Manchester United on Saturday and have now slipped into the Premier League relegation zone on goal difference following Sunderland's win on Sunday.
And the outlook appears to be bleak for Norwich as their last two fixtures are away to title contenders Chelsea and at home to Arsenal, who are in a battle for fourth place.
However, Snodgrass hopes the prestige of their opponents will help Norwich to raise their game.
"There is a massive amount of pressure that goes on, each day, as an individual, to give everything you have got," he said.
"You need to better yourself every day to give yourself the best possible chance to stay in this division.
"It doesn't get any easier, playing against the likes of Manchester United, Chelsea, Arsenal, but it's what you want to be in the Premier League for, to play against the best.
"We'll roll our sleeves up and try to give a good account of ourselves and get a few results."